<p>\(2.\) The continuous random variable \(X\) has probability density function given by</p>
<p>\[\mathrm{f}(x) = \begin{cases} k(2 - x), & \text{for}\hspace{0.5em} 0 \le x \le 2 \\ 0, & \text{otherwise} \end{cases}, \text{ where}\hspace{0.5em} k \text{ is a constant.}\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of \(k\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the cumulative distribution function of \(X\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">The continuous random variable \(Y\) is given by \(Y = 1 - \frac{1}{2}X\).</span></div><ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;\(\mathrm{P}(Y &lt; y) = y^2\), where&nbsp;\(0 \le y \le 1\).</div></li><li><div class="list-item-content"><span class="list-item-text">Deduce the probability density function of \(Y\) and hence, or otherwise, show that \[E(Y) = \frac{2}{3}.\]</span></div></li></ol></li></ol>

\(2.\) The continuous random variable \(X\) has probability density function given by

\[\mathrm{f}(x) = \begin{cases} k(2 - x), & \text{for } 0 \le x \le 2 \\ 0, & \text{otherwise} \end{cases}, \text{ where } k \text{ is a constant.}\]

Find the cumulative distribution function of \(X\).

The continuous random variable \(Y\) is given by \(Y = 1 - \frac{1}{2}X\).

Deduce the probability density function of \(Y\) and hence, or otherwise, show that \[E(Y) = \frac{2}{3}.\]

Finding a pdf from a cdf of a CRV

Finding a pdf from a cdf of a CRV